{"title":"The distinguishing index of connected graphs without pendant edges","authors":"W. Imrich, R. Kalinowski, M. Pilsniak, M. Wozniak","doi":"10.26493/1855-3974.1852.4f7","DOIUrl":null,"url":null,"abstract":"We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Math. Contemp.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.1852.4f7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 7
Abstract
We consider edge colourings, not necessarily proper. The distinguishing index D ′( G ) of a graph G is the least number of colours in an edge colouring that is preserved only by the identity automorphism. It is known that D ′( G ) ≤ Δ for every countable, connected graph G with finite maximum degree Δ except for three small cycles. We prove that D ′( G ) ≤ ⌈√Δ⌉ + 1 if additionally G does not have pendant edges.